To give an insight of the proposed method from the view of theoretical side, we formalize the problem as follows. We consider binary classification for simplicity. A domain is defined as a pair consisting of a distribution \(\mathcal {D}\)  on inputs \(\mathcal {X}\)  and a labeling function \(f: \mathcal {X}\rightarrow \mathcal {Y}\)  with \(\mathcal {Y}:= \lbrace 0, 1\rbrace \) . The labeling function can have a fractional (expected) value when labeling occurs non-deterministically. We denote by \(\langle \mathcal {D}_s, f_s\rangle \)  the source domain and \(\langle \mathcal {D}_t, f_t\rangle \)  the target domain. A hypothesis is a function \(h\in \mathcal {H}: \mathcal {X}\rightarrow \mathcal {Y}\) , where \(\mathcal {H}\)  is a hypothesis space on \(\mathcal {X}\)  with VC dimension [1]} \(d\) . The loss on source domain that a hypothesis \(h\)  disagrees with a labeling function \(f\)  is defined as
\(\mathcal {F}_s(h,f) := \mathrm {E}_{\mathbf {x}\sim \mathcal {D}_s}\left[|h(\mathbf {x})-f(\mathbf {x})| \right],\) 
